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```									                   Fibonacci Heaps

from CLRS, Chapter 20.

Princeton University • COS 423 • Theory of Algorithms • Spring 2002 • Kevin Wayne
Priority Queues

Heaps
Operation     Linked List   Binary   Binomial   Fibonacci †   Relaxed
make-heap           1         1         1             1         1
insert            1       log N     log N           1         1
find-min           N         1       log N           1         1
delete-min          N       log N     log N        log N       log N
union             1         N       log N           1         1
decrease-key         1       log N     log N           1         1
delete            N       log N     log N        log N       log N
is-empty           1         1         1             1         1

† amortized
this time
2
Fibonacci Heaps
Fibonacci heap history. Fredman and Tarjan (1986)
 Ingenious data structure and analysis.
   Original motivation: O(m + n log n) shortest path algorithm.
– also led to faster algorithms for MST, weighted bipartite matching
   Still ahead of its time.

Fibonacci heap intuition.
 Similar to binomial heaps, but less structured.
   Decrease-key and union run in O(1) time.
   "Lazy" unions.

3
Fibonacci Heaps: Structure
Fibonacci heap.
 Set of min-heap ordered trees.

min

17           24       23      7              3

30     26    46
marked   18   52    41

35
H                39         44
4
Fibonacci Heaps: Implementation
Implementation.
 Represent trees using left-child, right sibling pointers and circular,
–can quickly splice off subtrees
   Roots of trees connected with circular doubly linked list.
–fast union
   Pointer to root of tree with min element.
–   fast find-min

min

17             24   23         7                       3

30     26      46
18         52   41

35
H                      39              44
5
Fibonacci Heaps: Potential Function
Key quantities.
 Degree[x] = degree of node x.
   Mark[x] = mark of node x (black or gray).
   t(H) = # trees.
   m(H) = # marked nodes.
   (H) = t(H) + 2m(H) = potential function.

t(H) = 5, m(H) = 3
(H) = 11
degree = 3   min

17             24         23       7                   3

30      26     46
18   52    41

35
H                      39         44
6
Fibonacci Heaps: Insert
Insert.
 Create a new singleton tree.
   Add to left of min pointer.
   Update min pointer.

Insert 21

21
min

17            24           23    7                     3

30      26    46
18   52    41

35
H                     39         44
7
Fibonacci Heaps: Insert
Insert.
 Create a new singleton tree.
   Add to left of min pointer.
   Update min pointer.

Insert 21

min

17            24           23   7          21         3

30      26    46
18   52    41

35
H                    39         44
8
Fibonacci Heaps: Insert
Insert.
 Create a new singleton tree.
   Add to left of min pointer.
   Update min pointer.

Running time. O(1) amortized
Actual cost = O(1).                       Insert 21

   Change in potential = +1.
   Amortized cost = O(1).
min

17            24           23   7          21         3

30      26    46
18   52    41

35
H                    39         44
9
Fibonacci Heaps: Union
Union.
  Concatenate two Fibonacci heaps.
   Root lists are circular, doubly linked lists.

min                             min

23            24        17                    7         3         21

H'                                                                           H''
30      26    46
18   52    41

35
39         44
10
Fibonacci Heaps: Union
Union.
  Concatenate two Fibonacci heaps.
   Root lists are circular, doubly linked lists.

Running time. O(1) amortized
 Actual cost = O(1).
   Change in potential = 0.
   Amortized cost = O(1).

min

23            24         17                   7         3         21

H'                                                                           H''
30      26    46
18   52    41

35
39         44
11
Fibonacci Heaps: Delete Min
Delete min.
 Delete min and concatenate its children into root list.
   Consolidate trees so that no two roots have same degree.

min

7          24          23         17             3

30    26   46                           18      52      41

35                                39              44

12
Fibonacci Heaps: Delete Min
Delete min.
 Delete min and concatenate its children into root list.
   Consolidate trees so that no two roots have same degree.

current
min

7          24          23         17          18        52   41

30    26   46                                 39             44

35

13
Fibonacci Heaps: Delete Min
Delete min.
 Delete min and concatenate its children into root list.
   Consolidate trees so that no two roots have same degree.

0   1   2   3

current
min

7          24          23               17    18        52   41

30    26   46                                 39             44

35

14
Fibonacci Heaps: Delete Min
Delete min.
 Delete min and concatenate its children into root list.
   Consolidate trees so that no two roots have same degree.

0   1   2   3

current

min

7            24        23               17    18        52   41

30    26     46                               39             44

35

15
Fibonacci Heaps: Delete Min
Delete min.
 Delete min and concatenate its children into root list.
   Consolidate trees so that no two roots have same degree.

0   1   2   3

min

7          24          23               17    18        52   41

30    26   46                                 39             44
current

35

16
Fibonacci Heaps: Delete Min
Delete min.
 Delete min and concatenate its children into root list.
   Consolidate trees so that no two roots have same degree.

0   1   2   3

min

7          24          23               17          18       52       41

30    26   46                                       39                44
current

35
Merge 17 and 23 trees.

17
Fibonacci Heaps: Delete Min
Delete min.
 Delete min and concatenate its children into root list.
   Consolidate trees so that no two roots have same degree.

0   1   2   3

current
min

7          24                        17          18        52     41

30    26   46                        23          39               44

35
Merge 7 and 17 trees.

18
Fibonacci Heaps: Delete Min
Delete min.
 Delete min and concatenate its children into root list.
   Consolidate trees so that no two roots have same degree.

0   1    2   3

min           current

24                         7           18        52     41

26   46                17       30          39               44

35                     23
Merge 7 and 24 trees.

19
Fibonacci Heaps: Delete Min
Delete min.
 Delete min and concatenate its children into root list.
   Consolidate trees so that no two roots have same degree.

0   1    2   3

min           current

7          18    52   41

24       17       30         39         44

26    46       23

35
20
Fibonacci Heaps: Delete Min
Delete min.
 Delete min and concatenate its children into root list.
   Consolidate trees so that no two roots have same degree.

0   1    2   3

min           current

7      18        52   41

24       17       30     39             44

26    46       23

35
21
Fibonacci Heaps: Delete Min
Delete min.
 Delete min and concatenate its children into root list.
   Consolidate trees so that no two roots have same degree.

0   1    2   3

min                       current

7      18          52      41

24       17       30     39                  44

26    46       23

35
22
Fibonacci Heaps: Delete Min
Delete min.
 Delete min and concatenate its children into root list.
   Consolidate trees so that no two roots have same degree.

0   1    2   3

min                                    current

7           18       52         41

24       17       30          39                  44

26    46       23
Merge 41 and 18 trees.
35
23
Fibonacci Heaps: Delete Min
Delete min.
 Delete min and concatenate its children into root list.
   Consolidate trees so that no two roots have same degree.

0   1    2   3

min                                 current

7                52          18

24       17       30                    41     39

26    46       23                             44

35
24
Fibonacci Heaps: Delete Min
Delete min.
 Delete min and concatenate its children into root list.
   Consolidate trees so that no two roots have same degree.

0   1    2   3

min                                 current

7                52          18

24       17       30                    41     39

26    46       23                             44

35
25
Fibonacci Heaps: Delete Min
Delete min.
 Delete min and concatenate its children into root list.
   Consolidate trees so that no two roots have same degree.

min

7                    52        18

24     17    30                         41   39

26    46     23                               44
Stop.
35
26
Fibonacci Heaps: Delete Min Analysis
Notation.
 D(n) = max degree of any node in Fibonacci heap with n nodes.
   t(H) = # trees in heap H.
   (H) = t(H) + 2m(H).

Actual cost. O(D(n) + t(H))
 O(D(n)) work adding min's children into root list and updating min.
–at most D(n) children of min node
   O(D(n) + t(H)) work consolidating trees.
– work is proportional to size of root list since number of roots
decreases by one after each merging
–  D(n) + t(H) - 1 root nodes at beginning of consolidation

Amortized cost. O(D(n))
t(H')  D(n) + 1 since no two trees have same degree.
   (H)  D(n) + 1 - t(H).

27
Fibonacci Heaps: Delete Min Analysis
Is amortized cost of O(D(n)) good?

   Yes, if only Insert, Delete-min, and Union operations supported.
– in this case, Fibonacci heap contains only binomial trees since
we only merge trees of equal root degree
– this implies D(n)  log2 N

   Yes, if we support Decrease-key in clever way.
– we'll show that D(n)  log N, where  is golden ratio
– 2 = 1 + 
–  = (1 + 5) / 2 = 1.618…
– limiting ratio between successive Fibonacci numbers!

28
Fibonacci Heaps: Decrease Key
Decrease key of element x to k.
 Case 0: min-heap property not violated.
– decrease key of x to k
– change heap min pointer if necessary

min
7                    18         38

24     17     23      21     39         41

26     45
46     30             52

Decrease 46 to 45.
35    88     72
29
Fibonacci Heaps: Decrease Key
Decrease key of element x to k.
 Case 1: parent of x is unmarked.
– decrease key of x to k
– cut off link between x and its parent
– mark parent
– add tree rooted at x to root list, updating heap min pointer

min
7                     18         38

24     17      23      21     39         41

26     15
45     30              52

Decrease 45 to 15.
35      88     72
30
Fibonacci Heaps: Decrease Key
Decrease key of element x to k.
 Case 1: parent of x is unmarked.
– decrease key of x to k
– cut off link between x and its parent
– mark parent
– add tree rooted at x to root list, updating heap min pointer

min
7                     18         38

24     17      23      21     39         41

26     15     30              52

Decrease 45 to 15.
35      88     72
31
Fibonacci Heaps: Decrease Key
Decrease key of element x to k.
 Case 1: parent of x is unmarked.
– decrease key of x to k
– cut off link between x and its parent
– mark parent
– add tree rooted at x to root list, updating heap min pointer

min
15                                    7                     18         38

72                            24     17      23      21     39         41

26            30              52

Decrease 45 to 15.
35      88
32
Fibonacci Heaps: Decrease Key
Decrease key of element x to k.
Case 2: parent of x is marked.
– decrease key of x to k
– cut off link between x and its parent p[x], and add x to root list
– cut off link between p[x] and p[p[x]], add p[x] to root list
 If p[p[x]] unmarked, then mark it.
 If p[p[x]] marked, cut off p[p[x]], unmark, and repeat.

min
15                                     7                     18         38

72                             24     17      23      21     39         41

26             30              52

Decrease 35 to 5.
35
5      88
33
Fibonacci Heaps: Decrease Key
Decrease key of element x to k.
Case 2: parent of x is marked.
– decrease key of x to k
– cut off link between x and its parent p[x], and add x to root list
– cut off link between p[x] and p[p[x]], add p[x] to root list
 If p[p[x]] unmarked, then mark it.
 If p[p[x]] marked, cut off p[p[x]], unmark, and repeat.

min
15          5                          7                     18         38

72                             24     17      23      21     39         41

parent marked                26             30              52

Decrease 35 to 5.
88
34
Fibonacci Heaps: Decrease Key
Decrease key of element x to k.
Case 2: parent of x is marked.
– decrease key of x to k
– cut off link between x and its parent p[x], and add x to root list
– cut off link between p[x] and p[p[x]], add p[x] to root list
 If p[p[x]] unmarked, then mark it.
 If p[p[x]] marked, cut off p[p[x]], unmark, and repeat.

min
15          5       26                 7                     18         38

72                  88         24     17      23      21     39         41

30
parent marked                52

Decrease 35 to 5.
35
Fibonacci Heaps: Decrease Key
Decrease key of element x to k.
Case 2: parent of x is marked.
– decrease key of x to k
– cut off link between x and its parent p[x], and add x to root list
– cut off link between p[x] and p[p[x]], add p[x] to root list
 If p[p[x]] unmarked, then mark it.
 If p[p[x]] marked, cut off p[p[x]], unmark, and repeat.

min
15          5       26       24        7                     18         38

72                  88                17      23      21     39         41

30              52

Decrease 35 to 5.
36
Fibonacci Heaps: Decrease Key Analysis
Notation.
 t(H) = # trees in heap H.
   m(H) = # marked nodes in heap H.
   (H) = t(H) + 2m(H).

Actual cost. O(c)
 O(1) time for decrease key.
   O(1) time for each of c cascading cuts, plus reinserting in root list.

Amortized cost. O(1)
t(H') = t(H) + c
   m(H')  m(H) - c + 2
– each cascading cut unmarks a node
– last cascading cut could potentially mark a node
     c + 2(-c + 2) = 4 - c.

37
Fibonacci Heaps: Delete
Delete node x.
 Decrease key of x to -.
   Delete min element in heap.

Amortized cost. O(D(n))
O(1) for decrease-key.
   O(D(n)) for delete-min.
   D(n) = max degree of any node in Fibonacci heap.

38
Fibonacci Heaps: Bounding Max Degree
Definition. D(N) = max degree in Fibonacci heap with N nodes.
Key lemma. D(N)  log N, where  = (1 + 5) / 2.
Corollary. Delete and Delete-min take O(log N) amortized time.

Lemma. Let x be a node with degree k, and let y1, . . . , yk denote the
children of x in the order in which they were linked to x. Then:

0      if i  1
degree ( yi )  
 i  2 if i  1
Proof.
 When yi is linked to x, y1, . . . , yi-1 already linked to x,
 degree(x) = i - 1
 degree(yi) = i - 1 since we only link nodes of equal degree
   Since then, yi has lost at most one child
–otherwise it would have been cut from x
   Thus, degree(yi) = i - 1 or i - 2

39
Fibonacci Heaps: Bounding Max Degree
Key lemma. In a Fibonacci heap with N nodes, the maximum degree of
any node is at most log N, where  = (1 + 5) / 2.

Proof of key lemma.
 For any node x, we show that size(x)  degree(x) .
– size(x) = # node in subtree rooted at x
– taking base  logs, degree(x)  log (size(x))  log N.
   Let sk be min size of tree rooted at any degree k node.
– trivial to see that s0 = 1, s1 = 2
sk    size ( x*)
– sk monotonically increases with k                           k
   Let x* be a degree k node of size s k,               2   size( yi )
i 2
and let y1, . . . , yk be children in order                     k
that they were linked to x*.                         2   sdeg[ y ]i
i 2
k
 2   si  2
Assume k  2                        i 2
k 2
 2   si
i 0
40
Fibonacci Facts
1              if k  0

Definition. The Fibonacci sequence is:        Fk   2             if k  1
 F F          if k  2
 1, 2, 3, 5, 8, 13, 21, . . .                     k -1 k -2
• Slightly nonstandard definition.

Fact F1. Fk  k, where  = (1 + 5) / 2 = 1.618…

k 2
Fact F2.    For k  2, Fk  2   Fi
i 0           sk    size ( x*)
k
 2   size( yi )
i 2
Consequence. sk  Fk  k.                                         k
   This implies that size(x)  degree(x)               2   sdeg[ y ]i
i 2
for all nodes x.
k
 2   si  2
i 2
k 2
 2   si
i 0
41
Golden Ratio

Definition. The Fibonacci sequence is: 1, 2, 3, 5, 8, 13, 21, . . .
Definition. The golden ratio  = (1 + 5) / 2 = 1.618…
 Divide a rectangle into a square and smaller rectangle such that the
smaller rectangle has the same ratio as original one.

Parthenon, Athens Greece
42
Fibonacci Facts

43
Fibonacci Numbers and Nature

Pinecone

Cauliflower

44
Fibonacci Proofs

Fact F1. Fk  k.                    Fk  2    Fk  Fk 1
Proof. (by induction on k)
  k   k 1
Base cases:
  k (1   )


– F0 = 1, F1 = 2  .
   Inductive hypotheses:
  k ( 2 )            2 =  + 1
–   Fk  k and Fk+1  k+1               k 2

k 2
Fact F2. For k  2, Fk  2   Fi
i 0
Proof. (by induction on k)
 Base cases:                                Fk  2    Fk  Fk 1
k 2
– F2 = 3, F3 = 5                                  2   Fi  Fk 1
   Inductive hypotheses:                                     i 0
k
k 2                            2   Fk
Fk  2   Fi                                     i 0
i 0

45
On Complicated Algorithms
"Once you succeed in writing the programs for [these] complicated
algorithms, they usually run extremely fast. The computer doesn't
need to understand the algorithm, its task is only to run the
programs."

R. E. Tarjan

46

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